Factorial
A chain reaction of multiplication.
A number of the form n! = 1×2×…×n, like 6, 120 or 5040. The number of ways to line up n people — and one of the most explosive sequences around.
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Only 9 factorials live in the range: 1, 2, 6, 24, 120, 720, 5040, 40320 and 362880 — 10! = 3,628,800 overshoots, making factorials 0.0009% of the range, among the scarcest math features. Factorials explode because they count permutations: each extra person multiplies the arrangements by n.
Factorials and highly divisible numbers are natural allies: n! swallows every prime factor from 1 to n, so divisors pile up. From 6! = 720 (30 divisors) onward, every factorial in range is highly divisible, and 9! = 362880 boasts 160 divisors.
Trivia: 5040 = 7! has 60 divisors, and Plato picked it in the Laws as the ideal citizen count of a city-state precisely because it divides evenly so many ways. Also, 6! = 720 can be disguised as 8×9×10 — a factorial posing as three consecutive factors.